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Towards the end of the previous lecture we
started talking about issues related to model
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reduction and sub-structuring schemes. So
we'll continue with this discussion now, so
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we'll quickly recall where do we need you
know, model reduction and sub-structuring's.
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Actually, the need for using model reduction
and sub-structuring arises in several context,
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one is for example in treating very large
scale problems or in dealing with situations
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when the results from experiments need to
be discussed in congestion with prediction
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from mathematical models, that means for a
given structure we have made both finite element
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model as well as we have done some experimental
investigations and we want to now reconcile
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the two, and this is an essential step in
problems of finite element model updating,
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and in such situations the question of model
reduction arises, because of mismatched between
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degrees of freedom in measurement and computational
models, typically in a computational model
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the size of the, the number of degrees of
freedom can be very large and for every degree
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of freedom in the computational model we may
not have a sensor for instance it may not
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be feasible to measure degrees of freedom
at interior nodes in a 3-dimensional structure
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or measurement of rotations and so on and
so forth. So the number of degrees of freedom
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that we'll be able to measure in an experimental
work will typically be much less than the
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degrees of freedom in a finite element model.
Another situation is when different parts
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of a structure are developed by different
teams, possibly by using both experimental
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and computational tools, and based on this
we need to construct the model for built up
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structures, so this typically happens in applications
such as space applications and automotive
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applications, and this could also occur in
problems of secondary systems in civil engineering
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applications. There is a modern testing strategy
known as hybrid simulations, where we combine
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both experimental and computational models
for the same structure, so what we do is a
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part of the structure is studied experimentally
and a part of the structure computationally,
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and we want to in some way couple these two
disparate studies and arrive at certain conclusions
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on global behavior of the complete structure.
So let us start with discussion on problem
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of model reduction, so we will be limiting
our attention to linear time-invariant vibrating
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systems, so a typical finite element model
for a linear system will be of this form MX
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double dot + CX dot + KX = F(t) and certain
specified initial conditions, this is an end
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product of making finite element model as
you have seen in previous lectures.
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The objective of model reduction is to replace
this above this end degree of freedom system
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by an equivalent lowercase n degree of freedom
system where the reduced degree of freedom
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is much less than the capital N degrees of
freedom here, so for the reduced system the
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equation will be again of the form MR XM double
dot + CR XM dot + KR XM is FR(t), the subscript
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R here refers to reduced model, whereas the
subscript M, I will shortly come to that,
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this is the vector of degrees of freedom which
have been retained in the reduced model, so
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in the original model what we do is the degree
of freedom is partitioned into two sets, one
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set is called master degrees of freedom, the
other set is called slave degrees of freedom,
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so this subscript M here refers to the master
degrees of freedom which have been retained
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in the reduced model. The slave degrees of
freedom XS(t) have been eliminated from this
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model and a reduced model of this type has
been arrived at, so the size of the master
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degree of freedom will be lower case n cross
1, and slave degrees sorry, master degrees
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of freedom, slave degrees of freedom will
be N - N cross 1 vector, XS(t) is a N - N
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cross 1 vector.
Now in all, there are several methods for
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model reduction and in all the alternative
method there is a generic form to the problem
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of model reduction, so what we do is X(t)
is written as partitioned as already I mentioned
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as master and slave, and this we take it to
the, this X(t) is taken to be related to XM(t)
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through a transformation matrix capital Sai,
so XM is lowercase n cross 1, and capital
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Sai therefore will be M cross N transformation
matrix, so we can substitute this into the
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governing equation we get the equation at
this stage in this form, and if you pre multiply
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by sai transpose I get equation of this kind,
and I call this MR which is sai transpose
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M sai as a reduced mass matrix, clearly if
you take transpose of this it will be a since
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capital M is symmetric this MR would also
be symmetric. Similarly we define reduced
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damping matrix, and reduced stiffness matrix,
we call this quantity FR(t) as sai transpose
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F(t) as a reduced force vector, so once this
is achieved we can analyze this equation using
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any of the tools that we already developed,
but therefore the question now remains how
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do we select this transformation matrix?
Now there could be different criteria based
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on which we may like to select this transformation
matrix, for example the original model would
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have capital N number of natural frequencies
and mode shapes, on the other hand the reduced
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model will have only lowercase n Eigen pairs,
now suppose I have a 100 degrees of freedom
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system and I reduce it to a 10 degrees of
freedom system, the 100 degrees of freedom
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system will have 100 natural frequencies and
100/100 modal matrix whereas the reduced model
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will have 10 natural frequencies and 10/10
model matrix, these 10 natural frequencies
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of the reduced model should they be equal
to any of these 100 natural frequencies of
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the larger model, that could be one of the
criteria, or should the frequency response
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function over a given frequency range of the
reduced system serve as an acceptable approximation
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to the corresponding FRF's of the original
system, so here we are matching response,
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here we are matching only the natural frequency,
so if we match FRF's the issue is related
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to mode shapes as well as damping models would
be allowed for.
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Similarly should transient response to dynamic
excitation for the reduced system serve as
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an acceptable approximation to the response
of the original system, so we can set forth
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different objectives depending on which one
is of crucial importance to a given situation,
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we have to suitably design this transformation
matrix.
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Now before we proceed further we can just
as there is a model reduction, there is a
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counter you know feature that is model expansion,
for example consider a structural system that
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is being studied both experimentally and computationally,
so this n is a number of major degrees of
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freedom, whereas capital N is a degrees of
freedom in the computational model, and this
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typically far exceeds the measured degrees
of freedom in the experimental model. So now
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when we reconcile either I can reduce my computational
model to match the number of measured DOF's,
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so let me go back to again the example of
capital N being 100 and lowercase n being
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10, so what I can do is from a 100 degree
of freedom computational model I can illuminate
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90 degrees of freedom and obtain a model with
10 degrees of freedom and the degrees of freedom
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can be chosen to match what exactly I have
measured.
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On the other hand we could also expand the
measurement model, that means I have 10 degrees
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of freedom model here I will augment it by
additional 90 degrees of freedom, so that
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augmentation is essentially a transformation,
so if I do that then instead of calling it
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as model reduction, it would become model
expansion, because a smaller model is now
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replaced by a larger model, so the transformation
that we discuss can be viewed from both these
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perspectives. So that is to say reduce the
size of the computational model so that only
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the degrees of freedom which are common to
both experimental and computational models
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are retained, or alternatively expand the
size of the experimental model so that the
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degrees of freedom in both experimental and
computational model match.
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Now we will discuss 3 alternative techniques
for model reduction, and the names of these
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techniques are static condensation method,
dynamic condensation, and there is what is
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known as system equivalent reduction expansion
process. So I will just run through the logic
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of these 3 model reduction schemes and we
will discuss the relative merits and demerits.
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So-called static condensation is also called
Guyan's reduction technique, so what we are
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looking for? We have this global degrees of,
the degrees of freedom for the larger system
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partitioned as master and slaves, and I want
to now relate X(t) to the masters through
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this transformation matrix capital Sai, so
this partitioning of states into master and
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slave induces a partition on the structural
matrices and I can write the equation in this
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form, so here we assume that the slave degrees
of freedom do not carry any external force,
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okay, that's an assumption. Now the idea in
static condensation is to relate the master
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and slave degrees of freedom through a relation
which is valid only for under static conditions,
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that means to establish relationship between
XM and XS I will consider the equilibrium
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equation, this equation I will discard the
inertial and damping terms and write only
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the static equilibrium equation. So using
this equation now I will be able to establish
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a relationship between XM and XS, so if this
is what
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we are going to accept as a relationship between
master and slaves the first of this equation
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gives KMM XM + KMS XS is FM(t), that we are
not considering.
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The second equation is of interest to us KSM
into XM + KSS into XS is 0, so by rearranging
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the terms I get XS as - KSS inverse KSM XM,
so this is a relation between slave and master.
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So the transformation matrix therefore can
be written as I into, sorry I and - KSSs inverse
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KSN, so this will be lowercase n by n, this
will be capital N - N across N - N square
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matrix, so this is the sai matrix. Now we
can also look at now the expression for kinetic
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energy and potential energy in the original
system, the expression for kinetic energy
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is 1/2 X dot transpose MX dot, so X I'm writing
it as sai into XM, so if I make that substitution
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for X dot I will write sai XM dot and for
X dot transpose it will be XM dot transpose
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sai transpose. So if I now call this quantity
sai transpose M sai as MR, I get the expression
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for kinetic energy in the reduced model as
shown here, so this MR is now the reduced
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mass matrix.
Similarly the potential energy I can write
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1/2 X transpose KX so this again 1/2 sai XM
here, and XM transpose, sai transpose here,
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and I get a reduced stiffness matrix KR which
is sai transpose K sai. Now therefore the
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governing equation, I can now write for the
reduced system as MR XM double dot + CR XM
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dot + KR XM = FR(t), so this equation can
now be analyzed, this is the reduced model
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that we are looking for. So what are the features
of this? So we can
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work out the details of the MR and KR matrices
in a more explicit manner, so sai transpose
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M sai will give me this, this is sai transpose,
this is this, and if I expand this I will
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get now the reduced mass matrix in this form,
you must notice here that the reduced mass
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matrix is now a function of the stiffness
matrices of the original system, this is unusual
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because the inertial, kinetic energy in the
reduced system is now function of stiffness
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characteristics, okay, that's an artifice
induced because of the remodeled reduction
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that we have done. Similarly the reduced stiffness
matrix is sai transpose K sai and this by
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rearranging the terms I get this as the reduced
stiffness matrix.
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Okay, now we can make few observations, now
the slave degrees of freedom are related to
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the master degrees of freedom through relations
that are strictly valid for static situations,
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and hence this method is known as method of
static condensation. The partitioning of degrees
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of freedom as being masters and slaves has
to be done by analyst bearing in mind this
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assumption, that master and slaves are connected
to each other through relations which has
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strictly valid only under static conditions
so what is the consequence of that? The method
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is likely to perform better if slave degrees
of freedom contribute little to the kinetic
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energy, so in regions of low mass and high
stiffness, you should identify the slave degrees
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of freedom. The select slave degrees of freedom
such that the lowest Eigenvalue of the equation
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KS is alpha = lambda MSS alpha has the highest
Eigenvalue, that means between two competing
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choices for slave degrees of freedom you will
get two different KSS and MSS, you perform
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the Eigenvalue analysis for the two competing
choices, and between the two the one which
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has, the one in which the lowest Eigenvalue
is higher is a better choice, so I will illustrate
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that with an example, this again is a consequence
of the basic fact that we are only, they're
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relating the master and slave only through
static relations. So as I was telling select
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slave degrees of freedom in regions of high
stiffness and low mass.
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Now we to ensure that terms of MSS are small
in, and in terms of KSS are last that is I
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am reiterating the same statement in a slightly
different way, yet another way of saying similar
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thing is those degrees of freedom which yield
the larger value for this ratio can be selected
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as slaves, so you can alter the all the degrees
of freedom based on this ratio and select
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as many slaves as this you needed by assessing,
by comparing this ratio.
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Now it's clear that the error due to model
reduction increases with increases in driving
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frequencies of interest, that is because with
increase in driving frequencies the kinetic
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energy goes on increasing and we can't ignore
mass, a mass which is small at low frequency
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will contribute significantly to kinetic energy
at a higher frequency therefore the assumption
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will start breaking down. Now any initial
condition specified on slave degrees of freedom
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would not be satisfied, especially the velocity
degrees of freedom and things like, even displacement
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degrees of freedom because we're even slave
is made to you know forcefully related to
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the master and the initial condition of slave
cannot be accommodated in the further modeling
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work. The static condensation does not reproduce
any of the original natural frequencies of
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the original analytical model, and all the
natural frequencies of the reduced models
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would be higher than those of the full model.
Now again let me go back to the example of
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a bigger model with 100 degrees of freedom
and a reduced model of 10 degrees of freedom.
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The reduced model if you compute the 10 natural
frequencies for the reduced model, these frequencies
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need not agree with any of the 100 natural
frequencies of the original system, there
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is no guarantee that is bound to happen if
you follow this procedure.
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So we can try to understand this through a
numerical example, so let's consider a simple
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example having 6 degrees of freedom with springs
as shown here, and we'll assume stiffness
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is 1000 newton per meter, and mass is 10 kg,
so this is simple you know vibrating system,
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there was the only thing of interest is this
X1 and X6 are coupled so the stiffness matrix
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will have certain you know of diagonal terms
to reflect that, so this stiffness matrix,
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this is a stiffness
matrix for the system as you can see there
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is a term here which reflects coupling between
first and this, the sixth mass, this mass
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and this mass, this is a mass matrix as one
could expect, this is a diagonal matrix since
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we are using lumped mass matrix, modeling,
and you can do the Eigenvalue analysis, you
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can find the, if you perform that the modal
matrix comes out to be this and this is the
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vector of natural frequencies expressed in
radian per second, so let us say that this
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is my larger model.
Now I want to achieve model reduction, okay
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by using static condensation, so to begin
with I can compute this ratio K(I,I)/M(I,I),
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now the ratios are shown here so let us consider
for purpose of illustration, two alternative
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choices for master and slave degrees of freedom,
in each case we will have, we will try to
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reduce the model to a 3 degree of freedom
system, so in the first case what I will take
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the master degrees as 1, 2, 3 and slave degrees
as 4, 5, 6, now is it a good choice, is it
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a deliberately bad choice because we want
this ratio to be large for slaves, but I'm
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forcing it do you know what should be ideally
slave degrees of freedom, I am making them
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as masters just to emphasize what would happen.
Then I will perform this Eigenvalue analysis
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KSS alpha = lambda into alpha, and these are
the Eigenvalues.
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Now the sai matrix turns out to be this and
I can get the reduced mass and stiffness matrices
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and get the Eigenvalue, Eigenvector matrix
for the reduced system, and the 3 natural
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frequencies. Now the 3 natural frequencies
we are getting as 5.5, 19.7, and 22.07, so
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if you go here and see this is 4.6, 13, 13,
19, 22, and 29, so we don't seem to be, we
196
00:20:33,040 --> 00:20:39,370
seem to be getting 2 frequencies in this,
for these 2 frequencies seem to be giving
197
00:20:39,370 --> 00:20:46,970
reasonable answers but the first 3 modes are
not captured well.
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00:20:46,970 --> 00:20:56,390
Now let us know switch the options and I will
make now 4, 5, 6 as masters and 1, 2, 3 as
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slaves, this is what our recommendation you
know suggests, that this is what we should
200
00:21:01,590 --> 00:21:05,930
be doing if you are interested in producing
a 3 degree of freedom model. Now let's again
201
00:21:05,930 --> 00:21:13,830
do this Eigenvalue analysis and I get 290,
419 and 91, now the point I was making was
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00:21:13,830 --> 00:21:20,780
the lowest Eigenvalue here is 53.89, the lowest
Eigenvalue here is 290 so between the two
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00:21:20,780 --> 00:21:26,870
model the one which has higher lowest Eigenvalue
is the second model, because it is 290, this
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00:21:26,870 --> 00:21:31,830
290 is much larger than 53 so we could expect
that this will perform better. And in static
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00:21:31,830 --> 00:21:37,860
condensation the reduced model should typically
represent the behavior of the system in low
206
00:21:37,860 --> 00:21:45,330
frequencies well, okay. Now the reduced model
looks like this and I get the 3 natural frequencies
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00:21:45,330 --> 00:21:54,010
to be this, so we see here 4.6, 13.1, 15.14
so relatively speaking they seems to be you
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00:21:54,010 --> 00:22:02,890
know a better, you know reduction has been
you know obtained through this choice of master
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00:22:02,890 --> 00:22:08,789
and slaves. So at this stage we should notice
that in the reduced model the frequencies
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00:22:08,789 --> 00:22:16,559
that I obtain do not match with any of the
natural frequencies of the global model.
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00:22:16,559 --> 00:22:22,559
Now the next question I should ask is suppose
I demand, okay some frequency should match,
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00:22:22,559 --> 00:22:27,411
okay how to achieve that? So that takes us
to the discussion on what is known as dynamic
213
00:22:27,411 --> 00:22:33,400
condensation technique, so let's consider
for sake of discussion a harmonically driven
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00:22:33,400 --> 00:22:41,030
undamped system and this is our equation,
and this we call it as dynamic stiffness matrix,
215
00:22:41,030 --> 00:22:46,690
this is - omega square M + K this we encountered
earlier, so the equilibrium equation in the
216
00:22:46,690 --> 00:22:52,669
frequency domain is DX into F. So now what
I do is I will again partition X into XM and
217
00:22:52,669 --> 00:23:00,410
XS and related to, X is related to XM through
this matrix. Now this partitioning induces
218
00:23:00,410 --> 00:23:03,900
this partitioning of the dynamic stiffness
matrix also as shown here. Now what I do here
219
00:23:03,900 --> 00:23:11,210
is, I will again assume the slave degrees
of freedom are not driven, so I can use the
220
00:23:11,210 --> 00:23:19,380
second equation here which is DSM XM + DSS
XS = 0 from which I get XS to be this, there
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00:23:19,380 --> 00:23:26,090
is no approximation here, okay I am not throwing
out any term, so the transformation matrix
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00:23:26,090 --> 00:23:35,410
that I am looking for is given by this. Now
this omega, okay is now a parameter in your
223
00:23:35,410 --> 00:23:42,780
model reduction, okay, so you've to make a
choice for this omega, that is in addition
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00:23:42,780 --> 00:23:46,680
to making choices on which of the degrees
of freedom should be master and which should
225
00:23:46,680 --> 00:23:57,530
be slaves this reduction scheme also demands
that you should make a choice on omega, okay.
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00:23:57,530 --> 00:24:05,059
Now again we can do the same steps and obtain
the reduced mass and stiffness matrices as
227
00:24:05,059 --> 00:24:10,340
shown here, this is what I was saying in addition
to choosing slave and master DOF's here one
228
00:24:10,340 --> 00:24:15,371
also need to specify the frequency omega at
which the condensation has to be done. The
229
00:24:15,371 --> 00:24:20,500
method requires the determination of inverse
of this matrix see here this matrix needs
230
00:24:20,500 --> 00:24:25,520
to be inverted. Again let me point out one
more thing, the reduced mass matrix here,
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00:24:25,520 --> 00:24:33,350
and the reduced stiffness matrix here now
depend on the mass stiffness and the driving
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00:24:33,350 --> 00:24:40,940
frequency in the original system, so these
reduced matrices MR and KR you have to, they
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00:24:40,940 --> 00:24:47,460
are not amenable for a direct physical interpretation,
okay, this inversion of the matrix can be
234
00:24:47,460 --> 00:24:52,210
computationally demanding so
we could adopt some simplifications if they're
235
00:24:52,210 --> 00:24:56,990
necessary, and while doing so we will also
see what is the relationship between dynamic
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00:24:56,990 --> 00:25:01,500
condensation and static condensation. So if
we consider the inverse of this I will pull
237
00:25:01,500 --> 00:25:09,470
this KSS term outside, and I can write in
this form, and the inverse of product is,
238
00:25:09,470 --> 00:25:13,720
product of inverse in the reversed order,
so this becomes this, and if you use what
239
00:25:13,720 --> 00:25:20,720
is known as Neumann expansion, I can expand
this matrix in this form, it's a expansion
240
00:25:20,720 --> 00:25:24,400
with infinite number of terms, only few terms
are shown here.
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00:25:24,400 --> 00:25:30,220
So if I now omit this higher order terms in
omega and retain only the two terms, I get
242
00:25:30,220 --> 00:25:43,450
this as my matrix, and if I instead of inverting
the matrix now I can use this you know matrix,
243
00:25:43,450 --> 00:25:50,860
and this method is called the improved reduction
scheme, it's an improvement over static condensation
244
00:25:50,860 --> 00:25:57,160
method , and it avoids the inversion of this
matrix. If you are going to repeat this calculation
245
00:25:57,160 --> 00:26:05,159
for different values of omega then this is
a simpler approach, we can also of course
246
00:26:05,159 --> 00:26:08,590
do something
else we can formulate a Eigenvalue problem
247
00:26:08,590 --> 00:26:15,080
associated with KSS and MSS, and suppose if
I consider KSS alpha as lambda MSS alpha,
248
00:26:15,080 --> 00:26:20,600
and if phi is a matrix of eigenvectors and
capital Lambda be the diagonal matrix of Eigenvalues
249
00:26:20,600 --> 00:26:26,860
such that phi transpose MSS is phi, and this
is lambda, then if I consider the problem
250
00:26:26,860 --> 00:26:32,300
of inverting this I can consider the set of
these equations, and if I make a transformation
251
00:26:32,300 --> 00:26:39,409
Y = phi U, where phi is this matrix of Eigenvectors
and substitute here, and use these orthogonality
252
00:26:39,409 --> 00:26:45,120
relations I will be able to show that this
inverse is nothing but this, and this is a
253
00:26:45,120 --> 00:26:55,350
diagonal matrix so it does not require inversion.
So this phi can be computed, see this I am
254
00:26:55,350 --> 00:27:01,909
computing phi there is no omega here, so the
same phi can be used for different omegas
255
00:27:01,909 --> 00:27:06,020
so that is the idea which affords simplification
here.
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00:27:06,020 --> 00:27:12,080
Now we will return to the example that we
considered here the same example, I will again
257
00:27:12,080 --> 00:27:16,340
follow the same choices of degrees of, master
and slave degrees of freedom, but now what
258
00:27:16,340 --> 00:27:22,490
I will do is I have no additional choice to
make on driving frequencies, so I have made
259
00:27:22,490 --> 00:27:28,169
several choices, so I have made 7 choices,
in the first choice I take omega to be 4.61
260
00:27:28,169 --> 00:27:33,680
which happens to be the first natural frequency
of the system, and I make from a 7 degree
261
00:27:33,680 --> 00:27:42,200
of freedom system I get a 3 degree freedom
system, so that 4.61 turns out to be the one
262
00:27:42,200 --> 00:27:47,250
of the Eigenvalues of the reduce system, and
there are two more frequencies. If omega has
263
00:27:47,250 --> 00:27:53,340
taken a 13, 13 happens to be, 13.09 happens
to be one of the frequencies, the other two
264
00:27:53,340 --> 00:28:00,730
of course are not the natural frequency of
the system, so by selecting 13.71, I ensure
265
00:28:00,730 --> 00:28:07,070
that 13.71 is one of the natural frequencies
of my reduced model and so on and so forth.
266
00:28:07,070 --> 00:28:13,529
So in choice 1 this is what I get. In choice
2 that means
267
00:28:13,529 --> 00:28:18,450
choice of master and slave degrees of freedom
the same thing happens, but of course now
268
00:28:18,450 --> 00:28:24,220
the frequencies other than the one that are
in proximity of these numbers will be different
269
00:28:24,220 --> 00:28:34,340
from what was there in model one, so the choice
of omega do matter, and those natural frequencies
270
00:28:34,340 --> 00:28:40,090
which are close to omega are predicted well
that is the observation that we make here,
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00:28:40,090 --> 00:28:47,740
so this 4.6, 13 these are the frequencies
they are captured here, the 6 degree of freedom
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00:28:47,740 --> 00:28:54,299
not 7 degree of freedom, so there are 6 alternative
models and each one works well in the neighborhood
273
00:28:54,299 --> 00:28:57,981
of the frequency chosen, and that fact is
independent of choice of master and slave
274
00:28:57,981 --> 00:29:02,970
degrees of freedom, no matter which is master
which is slave the fact that at omega, if
275
00:29:02,970 --> 00:29:10,430
you select omega = 19.93 the reduced model
will have that as one of the frequencies in
276
00:29:10,430 --> 00:29:19,090
both the cases.
So we can make few comments now, so in dynamic
277
00:29:19,090 --> 00:29:24,010
condensation, in addition to choice of master
slave DOF's the choice of omega also matters,
278
00:29:24,010 --> 00:29:28,820
those natural frequencies which are close
to omega are predicted well. In a harmonic
279
00:29:28,820 --> 00:29:34,191
response analysis omega can be chosen to be
equal to the driving frequency, so if you
280
00:29:34,191 --> 00:29:43,799
are varying omega then computationally this
form of, you know the writing the transformation
281
00:29:43,799 --> 00:29:49,299
matrix in this form is advantageous, because
you have to do only one Eigenvalue analysis,
282
00:29:49,299 --> 00:29:58,690
and you need not have to invert the matrix,
this matrix for every omega.
283
00:29:58,690 --> 00:30:09,710
If the FRF's need to be traced over a frequency
range, for every value of driving frequency
284
00:30:09,710 --> 00:30:17,440
the condensation needs to be made separately.
If omega is taken to match with the driving
285
00:30:17,440 --> 00:30:22,090
frequency, that is the best option that you
would have, because at least at the frequency
286
00:30:22,090 --> 00:30:25,809
where you are driving that nearby natural
frequencies are captured correctly in the
287
00:30:25,809 --> 00:30:31,750
reduced model, this is expected to lead an
acceptable results if modes are well separated
288
00:30:31,750 --> 00:30:40,240
and damping is light, okay, that's very clear,
because once you make a choice of omega, and
289
00:30:40,240 --> 00:30:45,630
you are driving a system harmonically at that
frequency the response contribution is dominated
290
00:30:45,630 --> 00:30:53,230
by a single mode, then the method is likely
to work well even for first response analysis.
291
00:30:53,230 --> 00:31:00,400
Now we can ask this question in dynamic condensation
in the reduced model we are able to capture
292
00:31:00,400 --> 00:31:08,510
only one mode correctly, so the next logical
question that we can ask is can we retain
293
00:31:08,510 --> 00:31:13,760
a subset of the natural the frequencies of
the original system in the reduced model in
294
00:31:13,760 --> 00:31:19,909
an exact manner, okay, suppose in 100 degree
freedom system there will be 100 natural frequencies
295
00:31:19,909 --> 00:31:25,649
and if I am looking at reduced model with
10 degrees of freedom, the reduced model will
296
00:31:25,649 --> 00:31:31,279
have 10 natural frequencies, the question
I'm asking is can we select this 10, all these
297
00:31:31,279 --> 00:31:36,540
10 natural frequencies, can they be the natural
frequencies of the original system, need not
298
00:31:36,540 --> 00:31:43,220
be the first 10, it can be any 10 that you
can arbitrarily specify, so if you can do
299
00:31:43,220 --> 00:31:49,169
that then you are achieving something substantial,
but then you have to input lot of details
300
00:31:49,169 --> 00:31:56,230
into the reduction scheme, so such a scheme
indeed exist and that is known as system equivalent
301
00:31:56,230 --> 00:32:02,770
reduction expansion process, and abbreviated
as a SEREP, so I will be using the term SEREP,
302
00:32:02,770 --> 00:32:07,549
so the main features of this reduction scheme,
and also as the name indicates it's an expansion
303
00:32:07,549 --> 00:32:12,090
scheme as well, but we will focus on reduction
aspect of it, so it preserves a collection
304
00:32:12,090 --> 00:32:19,190
of normal modes during the reduction process.
Suppose we consider N degree of freedom model
305
00:32:19,190 --> 00:32:25,620
for a linear vibrating system, and let capital
Phi denote the N cross P modal matrix that
306
00:32:25,620 --> 00:32:32,030
include the first P modes, okay as before
we partition X into master and slave degrees
307
00:32:32,030 --> 00:32:38,679
of freedom, this partitioning on X induces
a partitioning on the partially known modal
308
00:32:38,679 --> 00:32:43,460
matrix as phi M and phi S, so what are the
different sizes here? This phi is N cross
309
00:32:43,460 --> 00:32:53,850
P, phi M will be lowercase n cross p, where
this n is the size of the master degrees of
310
00:32:53,850 --> 00:32:59,809
freedom, and p is the number of modes retained,
so in dynamic condensation P was 1, now P
311
00:32:59,809 --> 00:33:10,390
can be more. Phi S is N - N cross P, and also
when I say P it is not just the first P, by
312
00:33:10,390 --> 00:33:14,809
selecting the appropriate vectors in the modal
matrix I am also specifying which of the P
313
00:33:14,809 --> 00:33:22,940
modes I am looking at. Now we will assume
that N is greater than P that means the size
314
00:33:22,940 --> 00:33:29,820
of the reduced model is larger than the number
of modes that you have in the global model.
315
00:33:29,820 --> 00:33:40,770
Now we want to introduce an N cross 1 vector
of generalized coordinates Z(t) through this
316
00:33:40,770 --> 00:33:51,620
relation, okay, this is X is equal to some
modal matrix into Z, that part is fine, so
317
00:33:51,620 --> 00:33:54,820
this is the
relation. Now XM(t) is we can put it in the,
318
00:33:54,820 --> 00:34:03,440
you can expand this I get this, so from which
I can, we can solve for Z(t) using this equation.
319
00:34:03,440 --> 00:34:08,409
Now the number of unknowns and number of equations
in this case would not match, therefore I
320
00:34:08,409 --> 00:34:13,399
cannot use inverse directly I will have to
use what is known as pseudo inverse, I will
321
00:34:13,399 --> 00:34:18,639
just shortly explain what should of inverse
is, but if you, right now we will accept that
322
00:34:18,639 --> 00:34:23,609
there is an operation known as pseudo inverse
as indicated here, and this is this, that
323
00:34:23,609 --> 00:34:29,250
is phi M + is this, where plus indicates pseudo
inverse.
324
00:34:29,250 --> 00:34:35,629
Now if this is acceptable then X(t) can be
written in this form, that is phi M, phi S
325
00:34:35,629 --> 00:34:45,440
into this this, so the sai matrix which relates
X to XM is now given by this. Mind you this
326
00:34:45,440 --> 00:34:51,649
capital Phi here is the modal matrix of the
complete system, so before you do model reduction
327
00:34:51,649 --> 00:34:56,950
you should perform the Eigenvalue analysis
in the large system, okay otherwise you cannot
328
00:34:56,950 --> 00:35:02,009
use this method so the reduced mass matrix
and reduced stiffness matrix are obtained
329
00:35:02,009 --> 00:35:09,950
as shown here.
Now let me quickly describe what is pseudo
330
00:35:09,950 --> 00:35:18,099
inverse, it is not a thorough discussion but
it tells you what the idea is, so the motivation
331
00:35:18,099 --> 00:35:24,250
is suppose if you consider a linear algebraic
equation X1 + 5X2 = 1, so we have 1 equation
332
00:35:24,250 --> 00:35:29,390
and 2 unknowns, so there will be an infinity
of solutions, you draw this line any point
333
00:35:29,390 --> 00:35:37,700
lying on that line is a solution to this equation,
however if we decide to pick the point that
334
00:35:37,700 --> 00:35:45,520
is closest to the origin as the solution,
by that I mean so all points lying on this
335
00:35:45,520 --> 00:35:50,229
straight line is a solution, but if I decide
that I will take this point which is closest
336
00:35:50,229 --> 00:36:00,539
to the origin as the solution, this one, right?
Then I get a unique solution, okay.
337
00:36:00,539 --> 00:36:09,200
So let's consider what that means. So let
us consider AX = B, where A is N cross M,
338
00:36:09,200 --> 00:36:14,210
and X is M cross 1, and B is N cross 1, okay,
so this is a question. Now let us consider
339
00:36:14,210 --> 00:36:20,009
the KS where M is greater than N that is number
of unknowns is greater than number of equations.
340
00:36:20,009 --> 00:36:26,240
Now what we do is the solution that minimizes
the norm X that is a distance from the origin
341
00:36:26,240 --> 00:36:41,819
to the line, this is given by this, this distance,
so we can show that this is given by what
342
00:36:41,819 --> 00:36:47,880
is known as ARM, ARM is A transpose, AA transpose
inverse, so ARM is known as right pseudo inverse
343
00:36:47,880 --> 00:36:52,299
of A.
Now on the other hand if number of unknowns
344
00:36:52,299 --> 00:36:59,229
is less than number of equations then I can
find A solution X naught that minimizes this
345
00:36:59,229 --> 00:37:05,739
norm, the error in satisfying this equation
is minimized, and we can do the simple calculation
346
00:37:05,739 --> 00:37:11,410
and show that X naught that is the solution
in this case is given by ALM into B, where
347
00:37:11,410 --> 00:37:15,969
ALM is given by this, and this is known as
left pseudo inverse of A. So what does these
348
00:37:15,969 --> 00:37:20,459
things mean, what do these things mean? A
simple example
349
00:37:20,459 --> 00:37:26,760
suppose I consider a 4 cross 6 matrix A and
define B as pseudo inverse of A, so I will
350
00:37:26,760 --> 00:37:35,240
use these definitions and compute the pseudo
inverse, B is computed like this. Now if I
351
00:37:35,240 --> 00:37:40,959
multiply A and B I get an identity matrix,
so in that sense B is a pseudo inverse of
352
00:37:40,959 --> 00:37:46,470
A, although this matrix is not a square matrix
I am able to define another B matrix so that
353
00:37:46,470 --> 00:37:52,759
AB is an identity matrix, that is why it is
called a pseudo inverse.
354
00:37:52,759 --> 00:37:59,259
Now if this is 6 cross 4, instead of 4 cross
6 the pseudo inverse will be 4 cross 6, so
355
00:37:59,259 --> 00:38:06,210
AB in that case is again a diagonal matrix,
okay, so this is a notion of pseudo inverse
356
00:38:06,210 --> 00:38:12,959
that we are using in developing this SEREP
transformation.
357
00:38:12,959 --> 00:38:23,930
Now I have been mentioning, I am referring
to experimental you know models so it is better
358
00:38:23,930 --> 00:38:28,510
at this stage to you know understand what
is the difference between modeling in an experimental
359
00:38:28,510 --> 00:38:35,680
work and in a computational work, so in a
typical computational loop we start with the
360
00:38:35,680 --> 00:38:39,670
continuum, we discretize, and suppose we are
dealing with time invariant linear systems,
361
00:38:39,670 --> 00:38:45,859
we discretize and form the structural matrices
MCK and the load vector F and write this equilibrium
362
00:38:45,859 --> 00:38:52,150
equation with these specified initial conditions.
Then I will perform the Eigenvalue analysis
363
00:38:52,150 --> 00:39:01,410
and determine the natural frequencies, mode
shapes, modal participation factors, and modal
364
00:39:01,410 --> 00:39:08,479
damping ratios, so this analysis is known
as modal analysis, that is given the structural
365
00:39:08,479 --> 00:39:15,550
matrices how to find the natural frequencies,
modal matrix, damping ratios, and the participation
366
00:39:15,550 --> 00:39:22,999
factors, so this is the modal analysis in
a computational modeling approach, where we
367
00:39:22,999 --> 00:39:27,979
solve an Eigenvalue problem, once this is
known we have seen already how to compute
368
00:39:27,979 --> 00:39:33,019
the frequency response function or impulse
response function, and either use this algebraic
369
00:39:33,019 --> 00:39:37,829
relation or this convolution relation and
obtain the response either in time or in frequency
370
00:39:37,829 --> 00:39:42,509
domain, this we have seen, so here it is a
10 time it is a convolution, in frequency
371
00:39:42,509 --> 00:39:47,400
it is a multiplication, this is what we do
in a computational modeling.
372
00:39:47,400 --> 00:39:55,700
In an experimental work this loop is reversed,
we start by measuring the response, okay so
373
00:39:55,700 --> 00:40:01,719
the story is here we apply known excitation
to a test structure and measure the response,
374
00:40:01,719 --> 00:40:06,670
and what we measure we process and get the
matrix of impulse response function and frequency
375
00:40:06,670 --> 00:40:15,089
response function, from this we extract natural
frequencies, mode shapes, damping ratios and
376
00:40:15,089 --> 00:40:22,930
participation factor, this process of obtaining
the modal information from measured responses
377
00:40:22,930 --> 00:40:30,339
is known as experimental modal analysis, this
is in contrast to the modal analysis in computational
378
00:40:30,339 --> 00:40:36,099
modeling where we knowing the structural matrices
we perform an Eigenvalue analysis and find
379
00:40:36,099 --> 00:40:42,410
these quantities, and that we use to compute
the response, here we measure the response
380
00:40:42,410 --> 00:40:48,339
and we extract this information from these
measured responses, and from this we would
381
00:40:48,339 --> 00:40:54,069
like to construct models for the structure
that means mass, stiffness, damping matrices,
382
00:40:54,069 --> 00:41:01,430
and so on and so forth. So the loop is you
know the directions are reversed here, so
383
00:41:01,430 --> 00:41:06,079
there will be fundamental difficulty whenever
we use these two alternative approach to the
384
00:41:06,079 --> 00:41:15,449
same problem, and that is where this question
of modal reduction and expansion become crucial.
385
00:41:15,449 --> 00:41:23,249
Now let me return to the example of that 6
degree of freedom system, and now apply SEREP,
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00:41:23,249 --> 00:41:30,109
so let us retain now first 3 modes, so to
implement SEREP I have to again declare certain
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00:41:30,109 --> 00:41:34,349
degrees of freedom as masters, and certain
degrees of freedom as slaves, additionally
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00:41:34,349 --> 00:41:40,959
I should specify which are the modes that
I want to replicate in my reduced model, how
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00:41:40,959 --> 00:41:45,650
many of them? So what I am selecting is I
am taking 3 modes, and I am taking the first
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00:41:45,650 --> 00:41:53,839
3 modes so I can get phi M and phi S by partitioning
the modal matrix of the 6 degree of freedom
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00:41:53,839 --> 00:41:59,130
system that's what I have done, and this is
a transformation matrix, okay, using this
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00:41:59,130 --> 00:42:03,019
I will construct the reduced mass matrix and
reduce stiffness matrix and perform the Eigenvalue
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00:42:03,019 --> 00:42:09,009
analysis on the reduced system, it is a 3/3
system, so that system now has these 3 natural
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00:42:09,009 --> 00:42:16,549
frequencies 4.60, 13 point this and this.
Now if you look back these are exactly the
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00:42:16,549 --> 00:42:23,579
3 frequencies of the larger model, okay, so
there is no, precisely mathematically exactly
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00:42:23,579 --> 00:42:26,910
this, this is the reduced modal matrix, and
I get the
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00:42:26,910 --> 00:42:36,180
reduced structural matrices as shown here,
actually this is nothing but this is sai transpose
398
00:42:36,180 --> 00:42:45,240
M sai, this is reduced, this is sai transpose
K sai.
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00:42:45,240 --> 00:42:54,219
Similarly if I now declare 4, 5, 6 as master
and 1, 2, 3 as slaves, there will be certain
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00:42:54,219 --> 00:42:59,410
changes in my features of the reduced model,
but the 3 natural frequencies will be identical
401
00:42:59,410 --> 00:43:04,170
again, so this is phi M and phi S, this is
a transformation matrix, this is slightly
402
00:43:04,170 --> 00:43:09,499
now different from this transformation matrix.
So again I will perform the Eigenvalue analysis,
403
00:43:09,499 --> 00:43:15,289
no surprises, the first 3 natural frequency
is exactly matched, the reduced modal matrix
404
00:43:15,289 --> 00:43:21,490
of course is now different,
okay, and reduce structural matrices are obtained
405
00:43:21,490 --> 00:43:28,559
here and the interesting thing is this pair
of KR and MR, and this pair of KR and MR although
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00:43:28,559 --> 00:43:36,180
they are different they share the same Eigenvalues,
okay, so that is the achievement of this method.
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00:43:36,180 --> 00:43:42,249
So what are the features of this? To implement
SEREP the user need to specify the number
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00:43:42,249 --> 00:43:47,459
of modes to be retained, the more indices
which modes, and also the slaves and master
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00:43:47,459 --> 00:43:52,990
DOF's. The choice of normal modes to be included
in the reduced model is arbitrary, for example
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00:43:52,990 --> 00:43:58,640
in a 100 degree freedom system you want to
select 10 modes, you can select 1, 18, 32,
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00:43:58,640 --> 00:44:05,680
and 76 and so on and so forth, you need not
be first 10 nor they need to be in a cluster.
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00:44:05,680 --> 00:44:09,509
The scheme preserves the collection of normal
modes during reduction, whatever you are identified
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00:44:09,509 --> 00:44:16,010
as the natural frequencies you should be retained
in the reduced model they will be faithfully
414
00:44:16,010 --> 00:44:22,479
retained. The transformation matrix is deduced
from the modal matrix here, the modal matrix
415
00:44:22,479 --> 00:44:27,989
can be incomplete, you need not have a square
modal matrix even you can work with rectangular
416
00:44:27,989 --> 00:44:33,400
modal matrix, this is what would happen if
you do experimental modal analysis, in an
417
00:44:33,400 --> 00:44:41,380
experimental model analysis the modal matrix
is seldom square, it will be always you know
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00:44:41,380 --> 00:44:48,770
a rectangular matrix, so it will be incomplete.
So knowledge of K of course is needed, if
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00:44:48,770 --> 00:44:53,220
you are doing computationally because you
need to find the modal matrix, this could
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00:44:53,220 --> 00:44:59,260
be of value if modal matrix is obtained experimentally
that in which case K and M need not be known,
421
00:44:59,260 --> 00:45:04,859
phi can be directly measured experimental.
The natural frequencies of the reduced system
422
00:45:04,859 --> 00:45:09,180
matches with the full system, natural frequency
is irrespective of choice of master and slave
423
00:45:09,180 --> 00:45:14,239
degree of freedom. The method can be used
for model reduction or for modal expansion,
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00:45:14,239 --> 00:45:18,200
so I have not discussed what exactly happens
if you use it for modal expansion, but in
425
00:45:18,200 --> 00:45:27,880
principle it can be done.
Now we have now talked about modal reduction,
426
00:45:27,880 --> 00:45:34,079
the next topic that is related to this type
of questions is, questions or what are known
427
00:45:34,079 --> 00:45:42,730
as coupling techniques. Again here this is
a device to treat large complex structures,
428
00:45:42,730 --> 00:45:47,849
and the problem is large complex structure
require handling of large size matrices so
429
00:45:47,849 --> 00:45:54,890
can we do something about that, and similarly
in a manufactured product, parts of this structure
430
00:45:54,890 --> 00:46:01,200
could be modeled experimentally in parts computationally.
Now the question is how to develop model for
431
00:46:01,200 --> 00:46:10,269
built up structures based on models for these
sub structures. Now the
432
00:46:10,269 --> 00:46:16,069
coupling techniques answer these questions,
a good coupling technique needs to process
433
00:46:16,069 --> 00:46:22,579
some desirable features, it must be versatile
enough to accept data either from experiments
434
00:46:22,579 --> 00:46:26,930
or from FE model, say part of a structure
can be modeled experimentally and part of
435
00:46:26,930 --> 00:46:33,900
the structure computationally, the computational
modeler will be able to give structural matrices,
436
00:46:33,900 --> 00:46:40,549
mode shapes, natural frequencies, and so on
and so forth. An experimental is typically
437
00:46:40,549 --> 00:46:46,130
would be able to give FRF's impulse response
function and if model information is extracted
438
00:46:46,130 --> 00:46:52,959
you will be able to give the natural frequencies
mode shapes, damping ratios, participation
439
00:46:52,959 --> 00:46:58,509
factors which is experimentally measured,
but the experimentalist will have difficulty
440
00:46:58,509 --> 00:47:06,299
in specifying the structural matrices.
Now each component can be treated by an accurate
441
00:47:06,299 --> 00:47:13,440
and refined model, you can use each component,
each sub-structure can be modeled with any
442
00:47:13,440 --> 00:47:18,740
level of refinement and you know detailed
modeling, any level of detail can be included
443
00:47:18,740 --> 00:47:23,540
in a model, components may have to be broken
into small enough subsystems which permit
444
00:47:23,540 --> 00:47:28,059
suitable experimental test or analytical modeling
to be carried out, that means the substructure
445
00:47:28,059 --> 00:47:35,049
a scheme should not constrain the user in
terms of you know, if user wishes to do this
446
00:47:35,049 --> 00:47:39,729
it should not be a constrained, any structural
modification which has to be applied at any
447
00:47:39,729 --> 00:47:44,869
time only involves the reanalysis of the affected
part, suppose there are A, B, C are three
448
00:47:44,869 --> 00:47:51,660
substructures, and if A is modified then we
should not end up analyzing B and C, okay,
449
00:47:51,660 --> 00:47:55,259
then the technique must permit analysis of
different components at different times and
450
00:47:55,259 --> 00:48:00,350
by different teams, this is what typically
happens in a you know space structures, or
451
00:48:00,350 --> 00:48:06,299
automotive systems, and so on and so forth,
and even you know mechanical systems in civil
452
00:48:06,299 --> 00:48:13,130
engineering applications like a turbine, or
piping, in an industrial structure, okay,
453
00:48:13,130 --> 00:48:19,229
so the different people will be doing different
products.
454
00:48:19,229 --> 00:48:25,650
Now typically what are the steps involved
in this? The steps involved are, we partition
455
00:48:25,650 --> 00:48:30,589
the physical system into number of sub-structures
with a proper choice of connection and interior
456
00:48:30,589 --> 00:48:35,430
coordinates, we need to decide upon the method
of analysis for each of the sub-structure
457
00:48:35,430 --> 00:48:39,670
that means the analytical or experimental,
we have to derive the respective subsystem
458
00:48:39,670 --> 00:48:45,519
models either by a theoretical or experimental
approach, then we need to carry out condensation
459
00:48:45,519 --> 00:48:50,180
of degrees of freedom at the subsystem level,
and we need to assess the effect of neglect
460
00:48:50,180 --> 00:48:55,260
of certain modes and coordinates.
Next, we need to formulate the subsystem equation
461
00:48:55,260 --> 00:49:02,140
of motion either using spatial coordinates
or modal coordinates, analysis of one substructure
462
00:49:02,140 --> 00:49:07,720
should not require the knowledge of dynamical
properties of remaining components, then we
463
00:49:07,720 --> 00:49:12,619
arrive at the reduced order equations for
the global structure by invoking interface
464
00:49:12,619 --> 00:49:19,229
displacement established for different component
models, okay, so this is what will lead to
465
00:49:19,229 --> 00:49:26,410
the coupled system. So the coupling techniques
can be
466
00:49:26,410 --> 00:49:34,769
classified based on how you model the subsystems,
for example for modeling the subsystems we
467
00:49:34,769 --> 00:49:39,039
could use structural matrices that is mass,
stiffness, and damping, and special coordinate
468
00:49:39,039 --> 00:49:44,630
like displacement degrees of freedom and so
on and so forth, or you can model each subsystem
469
00:49:44,630 --> 00:49:49,029
in terms of a set of natural frequencies,
mode shapes, damping ratios, and participation
470
00:49:49,029 --> 00:49:55,319
factors, both are equivalent, so depending
on how you choose we can have different types
471
00:49:55,319 --> 00:50:02,260
of coupling techniques, in one of the scheme
of classification, we classify this coupling
472
00:50:02,260 --> 00:50:06,289
techniques as the impedance complete techniques
and modal coupling techniques.
473
00:50:06,289 --> 00:50:11,630
In impedance coupling techniques reduction
within the substructure is performed in terms
474
00:50:11,630 --> 00:50:16,449
of spatial coordinates or with the help of
frequency response functions of the subsystems,
475
00:50:16,449 --> 00:50:22,819
we don't use modal information. In modal coupling
techniques reduced model for the subsystems
476
00:50:22,819 --> 00:50:28,630
are obtained in terms of subsystem normal
modes, so we need to develop the formulary
477
00:50:28,630 --> 00:50:39,759
for dealing with you know these coupling techniques
and will take up these questions in the next
478
00:50:39,759 --> 00:50:44,839
class, and specifically we will be talking
about method known as component mode synthesis
479
00:50:44,839 --> 00:50:49,549
which is widely used in practice. So with
this will conclude the present lecture.